(ed.). Gravitational waves (IOP, 2001)(422s).

326 Elementary introduction to pre-big bang cosmologyso that the dilaton is growing ifd + √ d + n > n. (16.164)For n = 6, in particular, this condition requires d > 3. This could represent apotential difficulty for the pre-big bang scenario, which might be solved, however,by quantum cosmology effects [87].The scale factor duality of the action (16.147) is, in general, broken by theaddition of a non-trivial dilaton potential (unless the potential depends on thedilaton through φ, of course). When the antisymmetric tensor B µν is includedin the action, however, the scale factor duality can be lifted to a larger group ofglobal symmetry transformations. To illustrate this important aspect of the stringcosmology equations, we will consider here a set of cosmological backgroundfields {φ,g µν , B µν }, for which a synchronous frame exists where g 00 = 1,g 0i = 0, B 0µ = 0, and all the components φ,g ij , B ij do not depend on thespatial coordinates.Let us write the actionS =− 1 ∫2λ d−1 sd d+1 x √ |g|e −φ [R + (∇φ) 2 − 112 H 2 µνα](16.165)directly in the synchronous gauge, as we are not interested in the field equations,but only in the symmetries of the action. We set g ij =−γ ij and we find, in thisgauge,whereƔ ij 0 = 1 2 ˙γ ij,Ɣ 0i j = 1 2 g jk ġ ik = 1 2 (g−1 ġ) i j = (γ −1 ˙γ) ijR 0 0 =− 1 4 Tr(γ −1 ˙γ) 2 − 1 2 Tr(γ −1 ¨γ)− 1 2 Tr( ˙γ −1 ˙γ),R j i =− 1 2 (γ −1 ¨γ) j i − 1 4 (γ −1 ˙γ) j i Tr(γ −1 ˙γ)+ 1 2 (γ −1 ˙γγ −1 ˙γ) j i , (16.166)Tr(γ −1 ˙γ)= (γ −1 ) ij ˙γ ji = g ij ġ ji , (16.167)and so on (note also that ˙γ −1 means (γ −1 )˙).antisymmetric tensor,Similarly we find, for theH 0ij = Ḃ ij , H 0ij = g ik g jl Ḃ kl = (γ −1 Ḃγ −1 ) ij ,H µνα H µνα = 3H 0ij H 0ij =−3Tr(γ −1 Ḃ) 2 . (16.168)Let us introduce the shifted dilaton, by absorbing into φ the spatial volume, asbefore:√| det gij |e −φ = e −φ , (16.169)from whichφ˙= ˙φ − 1 d2 dt ln(det γ)= ˙φ − 1 2 Tr(γ −1 ˙γ). (16.170)